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The Best-Fit Line. Linear Regression. How do you determine the best-fit line through data points?. Fortunately technology, such as the graphing calculator and Excel, can do a better job than your eye and a ruler!. y-variable. x-variable. The Equation of a Straight Line. y = mx + b

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the best fit line

The Best-Fit Line

Linear Regression

how do you determine the best fit line through data points
How do you determine the best-fit line through data points?

Fortunately technology,

such as the graphing calculator

and Excel, can do a better

job than your eye and a ruler!

y-variable

x-variable

slide3

The Equation of a Straight Line

y = mx + b

where m is the slope or Dy/Dx and b is the y-intercept

In some physical settings, b = 0 so the equation simplifies to:

y = mx

slide4

Linear regression minimizes

the sum of the squared deviations

y = mx + b

y-variable

deviation = residual

= ydata point – yequation

x-variable

linear regression
Linear Regression
  • Minimizes the sum of the square of the deviations for all the points and the best-fit line
  • Judge the goodness of fit with r2
  • r2 x100 tells you the percent of the variation of the y-variable that is explained by the variation of the x-variable (a perfect fit has r2 = 1)
slide6

r2 is low

r2 is high

Goodness of Fit: Using r2

y-variable

How about the value of r2?

x-variable

slide7

Strong direct relationship

99.1% of the y-variation is due to

the variation of the x-variable

slide8

Noisy indirect relationship

Only 82% of the y-variation is due to

the variation of the x-variable - what

is the other 18% caused by?

slide9

When there is no trend!

No relationship!

in excel
In Excel
  • When the chart is active, go to chart, and select Add Trendline, choose the type and on option select display equation and display r2
  • For calibration curves, select the set intercept = 0 option Does this make physical sense?
slide11

Does the set intercept = 0 option

make a difference?

Using the set intercept = 0 option

lowers the r2 value by a small amount

and changes the slope slightly

the equation becomes
The equation becomes

A = mc

or

A = 0.89c

99.1% of the variation of the absorbance is due to the variation of the concentration.